On relationship between contact surface rigidity and harmonic generation behavior in composite materials with mechanical nonlinearity at fiber-matrix interface (Singapore November 2017) N. Matsuda, K. Kawahara, M. Hojo and M. Nishikawa Graduate School of Engineering, Kyoto University, Kyoto daigaku-katsura, Nishikyo-ku, Kyoto, Japan. E-mail: matsuda@me.kyoto-u.ac.jp Abstract The fiber reinforced composite material has extremely complicated reflection/transmission characteristics of elastic waves. For this reason, it has been difficult to obtain information on the interface by the conventional ultrasonic method. On the other hand, nonlinear mechanical properties occur due to clapping between the interfaces and the changes of the real contact area on kissing bonds in adhesive joints. Therefore, it has been expected that evaluation of harmonic generation behavior makes it possible to evaluate the kissing bonds with high sensitivity. In this study, we formulate elastic wave scattering problem by eigenfunction expansion considering the nonlinear mechanical properties of the cylindrical interfaces. The interfaces of the fibers and the matrix have been modelled by springs with mechanical nonlinearity. Furthermore, we dealt with a nonlinear scattering problem on a stainless steel fiber in epoxy resin matrix as an example of the calculation. The analysis results have been discussed with respect to the backscatter amplitude of fundamental wave and second harmonics in the light of the resonance scattering. The backscatter amplitude of the second harmonics is decreased due to not only the resonance of the second harmonics but also the influence of the fundamental wave resonance. In addition, the backscatter amplitude of the second harmonics strongly depends on the contact surface rigidity. Keywords: Elastic wave, Cylindrical inclusion, Second-harmonic generation, Contact acoustic nonlinearity, Imperfect interface 1 Background It has been known that a resonance scattering occurs when elastic waves of specific frequencies scatter on a cylinder. Numerous researches of elastic wave scattering on a cylinder and a sphere have been conducted since application to practical non-destructive evaluation has been expected [1,2]. Moreover, resonant scattering on a cylinder which occurs at a specific frequency has, particularly, attracted researchers' interest [1-7]. Resonance scattering is a phenomenon in which the amplitude of the backscattered wave takes local minimums at specific frequencies. According to the Resonance Scattering Theory (RST), the resonance has been explained as the interference among the surface waves encircling the cylinder. It has been expected that evaluation of harmonic generation behavior makes it possible to evaluate imperfectly bonded surfaces with high sensitivity. By measuring reflection and scattering signals of ultrasonic waves, it is possible to evaluate the presence or absence of defects, if the defects and surrounding material have difference in the acoustic impedance. However, the reflection characteristics at the contact interfaces or the weakly bonded interfaces, which contact with residual [ID164] 1
stress, do not change as much as the reflection characteristics at rigidly bonded interfaces. On the other hand, nonlinear mechanical properties occur due to clapping between the interfaces and the change of the real contact area on kissing bonds in adhesive joints [8-10]. This type of nonlinearity has been known as Contact Acoustic Nonlinearity (CAN). It has been also known that the mechanical nonlinearity of the interface varies greatly depending on its coupling state [8,9]. While only the incident wave frequency is detected in the scattered wave on the interfaces without mechanical nonlinearity, the higher harmonics whose frequencies are integer multiples of the excited fundamental frequency are generated on the interface with mechanical nonlinearity. For this reason, many studies have been conducted to evaluate the soundness of interfaces by higher harmonics caused by the nonlinearity. However, harmonic generation behavior on the cylindrical interfaces, such as fibers, has been discussed in only few studies [11]. In this paper, we formulate the multiple scattering problem by eigenfunction expansion considering the nonlinear mechanical properties of the cylindrical interfaces and show an example of numerical analysis. The nonlinear spring interfaces [10], which can take into account the interface roughness, has been applied to the fiber-matrix interfaces model. Although this model can be numerically analyzed using the boundary element method [11], we formulate the problem using the eigenfunction expansion [12]. This method allows us to deal the multiple scattering problem with large number of fibers, such as wave scattering problem in fiber-reinforced composites. In order to consider the nonlinear spring interfaces, we also derive the boundary condition of the nonlinear component using the perturbation analysis, and propose a practical equation to calculate the nonlinear term. Note that the formulation of a nonlinear spring interface model by eigenfunction expansion has not been presented to the knowledge of the authors. Since the harmonic generation and scattering behavior in the composite material, which has many fibers, becomes very complicated, the fundamental and the second-harmonic wavefield were calculated in the case where a plane wave is incident on a single fiber as an example of the numerical analysis. The analysis results have been also discussed with respect to the backscatter amplitude of fundamental wave and second harmonics in the light of the resonance scattering. 2 Formulation of the Problem 2.1 Numerical Analysis Model In this paper, we consider a model where isotropic fibers (mass density ρ 2, Lamé constants λ 2, μ 2) having a radius a oriented in the x 3 direction are arranged in a isotropic matrix (mass density ρ 1, Lamé constant λ 1, μ 1) which occupies infinite space (see Fig. 1). Assuming that the displacement field is a plane strain state at the x 1- x 2 plane, the wavefield of this problem is treated as a two-dimensional [ID164] 2
wavefield. We also consider the stationary wavefield where a monochromatic wave (frequency f, angular frequency ω=2πf ) is incident on this model. The boundary conditions between the fibers and the matrix are given by the following nonlinear relationships. σσ int rrrr = KK rr (1 ββδuu rr )Δuu rr, (1) σσ int rrrr = KK θθ Δuu θθ, (2) int int where σσ rrrr and σσ rrrr represent the normal stress and the shear stress on the fiber-matrix interfaces, respectively, and Δuu rr and Δuu θθ indicate the differences of displacements between the fiber and the matrix. K r and K θ are contact surface rigidities in the radial direction and the circumferential direction, which are the adhesion state indices of the interfaces. The degree of the nonlinearity is denoted by β. Assuming that the boundary condition is the linear expression obtained by removing the second term in Eq. 1, a linear solution is derived. Using this linear solution, the solution of the nonlinear problem can be decomposed into a linear component and a nonlinear component. Henceforth in this paper, the physical quantities corresponding to the second-harmonic component are represented by adding 2f to the left hand corner (Ex. 2f uu rr or 2f σσ int rrrr ). In addition, those of the fundamental wave component are similarly denoted by adding f to the left hand corner. When the nonlinear component is sufficiently smaller than the linear component, the second-harmonic component of the boundary condition Eq.1 is expressed as, 2f σσ int rrrr = KK rr Δ 2f uu rr KK rr ββ Δ f uu rr 2. (3) Matrix Fiber Young s modulus E [GPa] 5.32 243 Poisson ratio ν 0.331 0.300 Density ρ [kg/m 3 ] 1.12 10 3 9.45 10 3 Figure 1: Geometry of the fibers and matrix. Table 1 Material parameters of matrix and fiber used in numerical analysis [ID164] 3
2.2 Governing Equations and Numerical Implementation According to the Helmholtz decomposition, the displacement fields of the fundamental and the second-harmonic wavefield can be expressed by the potential functions. f φφ,. f ψψ,. 2f φφ and. 2f ψψ. Each potential satisfies the Helmholtz equation. 2 + kk Lαα 2 f φφ = 0, 2 + kk Tαα 2 f ψψ = 0, { 2 + (2kk Lαα ) 2 } 2f φφ = 0, { 2 + (2kk Tαα ) 2 } 2f ψψ = 0, (4) where the subscripts α denote the matrix (α= 1) or the fiber (α = 2), and k Lα and k Tα indicate the wave numbers of the longitudinal wave and the transverse wave, respectively. The general solution of the above equation is expressed by the linear sum of Hankel functions or Bessel functions. Also, the potentials corresponding to excitation waves to each fiber, scattered waves from each fiber and refraction waves in each fiber can be represented by infinite sums of Hankel functions and Bessel functions using expansion coefficients. From the linear boundary conditions, relational expressions between the expansion coefficients of the fundamental wave are derived (details written in [12,13]). Using these expressions, simultaneous equations are obtained from the conditions on the excitation potentials of each fiber. In order to determine the expansion coefficients by numerical means, we limit the order of eigenfunction expansion to -n * to n *, since these simultaneous equations contain an infinite number of the expansion coefficients. By solving the simultaneous equations according to collocation method, we can evaluate the expansion coefficients of the fundamental wave. The driving force term of the second harmonics in Eq. 3, KK rr ββ Δ f uu rr 2, is determined by the difference of the fundamental wave displacement between the fiber and the matrix, Δ f uu rr. Consequently, x n, which is rewritten term of the driving force term (see [13]), is obtained from the expansion coefficients of the fundamental wave. In particular, xx jj nn = 1 2 KK rrββ PP jj jj mm PP nn mm mm=, (5) where PP nn jj is, PP jj nn = kk L1 HH (1) f jj nn (kk L1 aa) AA nn + iiii aa HH (1) f jj nn (kk T1 aa) BB nn + kk L1 JJ f jj nn (kk L1 aa) EE nn + iiii aa JJ f jj nn(kk T1 aa) FF nn kk L2 JJ f jj nn (kk L2 aa) CC nn + iiii aa JJ f jj nn(kk T2 aa) DD nn, (6) f jj where AA nn f jj, BB nn f jj,, FF nn are the expansion coefficients of the fundamental wave on the j th fiber. By substituting x n to the boundary condition of the second harmonics, the relational expressions of the expansion coefficients of the second harmonics can be derived. As in the case of obtaining the expansion [ID164] 4
coefficients of the fundamental wave, the expansion coefficients of the second harmonics are evaluated by the collocation method. Note that the upper limit of the eigenfunction expansion order of the second harmonics is taken to be 2n *, since the wavelength of the second harmonics is shorter than that of the fundamental wave. 3 Numerical Analysis 3.1 Numerical analysis parameters Since the harmonic generation behavior in the fiber reinforced composites, which has many fibers, becomes very complicated, the fundamental and the second-harmonic wavefield were calculated in the case where a plane wave is incident on a single fiber as an example of the numerical analysis. The fiber and the matrix have been assumed to be a stainless steel fiber of 370 μm radius and epoxy resin, respectively. The material properties of the fiber and the matrix are shown in Table 1. The incident wave has been assumed to be a longitudinal plane wave with the displacement amplitude of 10 nm propagating in x 1 direction. The maximum order in eigenfunction expansion has been set to n * = 8. 3.2 Results Figure 2 shows the frequency dependence of the backscatter amplitude of the fundamental wave and that of the second harmonics. The amplitude of the backscattered wave has been evaluated by the amplitude of the average vertical stress on the plane perpendicular to the x 1 axis at the location L = 6.6 mm in front of the center of the fiber. The backscatter amplitude evaluated here contains only the contribution of the scattered waves, and the stress caused by the incident wave is excluded. It should be noted that the horizontal axes of Figs. 2(a) and 2(b) are the non-dimensional frequency of the fundamental wave and that of the second harmonics, respectively. For example, when the non- Figure 2: Frequency dependence of the backscatter amplitude of (a) fundamental wave and that of (b) second harmonics. [ID164] 5
dimensional incident wave frequency ff is 6, we need to read the backscatter amplitude of the fundamental wave from the point corresponding to ff = 6 in Fig 2(a) and the backscatter amplitude of the second harmonics from the point corresponding to 2ff = 12 in Fig 2(b), respectively. Each line in Fig. 2 shows the results when the non-dimensional contact surface rigidity K r is 0.1, 1, 10 or 100. The matrix and the fiber are barely adhered in the case of K r = 0.1, and are nearly perfectly adhered in the case of K r = 100. 3.3 Discussions If the contact surface rigidity is large, then the backscatter amplitude of the fundamental wave takes minimum values when the non-dimensional frequency ff is 5.7 or 9.2. On the other hand, these valleys disappear as the contact surface rigidity decreases. This indicates that the valleys are caused by the resonance scattering for the large contact surface rigidity K r, and that the valleys disappear because of not being satisfied the resonance condition when the interface rigidity becomes small. The backscatter amplitude of the second harmonics also takes minimum values around 2ff = 5.7 or 9.2 similarly to the fundamental wave, which corresponds to the case where the second harmonics satisfy the resonance condition on the fiber. It should be emphasized that there are also local minimums at frequencies 2ff = 11.2 and 17.4, which corresponds to the case where the fundamental wave satisfies the resonance condition. These results indicate that the backscatter amplitude of the second harmonics is decreased due to not only the resonance of the second harmonics but also the influence of the fundamental wave resonance. In addition, Fig. 2(b) reveals that the backscatter amplitude of the second harmonics strongly depends on the contact surface rigidity. Remarkably, the amplitude is small when the interface is barely adhered or when the interface is completely adhered. It is reasonable to suppose that the backscatter amplitude increases because the strain energy stored by the nonlinear springs at the interface becomes large when the non-dimensional contact surface rigidity is about 1 to 10. Although this calculation is only one example, these results show the possibility of evaluating the contact surface rigidity using the backscatter amplitude of second harmonics. 4 Summary In this paper, we formulated elastic wave scattering problem by eigenfunction expansion considering the nonlinear mechanical properties of the cylindrical interfaces. The interfaces of the fibers and the matrix have been modeled by springs with mechanical nonlinearity. As an example of the calculation, we dealt with a nonlinear scattering problem on a stainless steel fiber in an epoxy resin matrix. The analysis results suggest that the backscatter amplitude of the second harmonics is decreased due to not [ID164] 6
only the resonance of the second harmonics but also the influence of resonance of the fundamental wave. In addition, the backscatter amplitude of the second harmonics strongly depends on the contact surface rigidity. Especially the backscatter amplitude increases when the non-dimensional contact surface rigidity is about 1 to 10. References [1] J. J. Faran, Sound Scattering by Solid Cylinders and Spheres, J. Acoust. Soc. Am., vol. 23, no. 4, pp. 405-418, Jul., 1951. [2] N. D. Veksler, Scattering of an obliquely incident plane acoustic wave in Resonance Acoustic Spectroscopy, Berlin, Springer-Verlag, 1993, pp. 1-15. [3] L. Flax, L. R. Dragonette, and H. Überall, Theory of Elastic Resonance Excitation by Sound Scattering, J. Acoust. Soc. Am., vol. 63, no. 3, pp. 723-731, Mar., 1978. [4] F. Honarvar and A. N. Sinclair, Acoustic Wave Scattering from Transversely Isotropic Cylinders, J. Acoust. Soc. Am., vol. 100, no. 1, pp. 57-63, Jul., 1996. [5] Y. Fan et al., Circumferential Resonance Modes of Solid Elastic Cylinders Excited by Obliquely Incident Acoustic Waves, J. Acoust. Soc. Am., vol 113, no. 1, pp. 102-113, Jan., 2003. [6] F. Honarvar, Y. Fan, and A. N. Sinclair, Scattering of Acoustic Waves from Immersed Transversely Isotropic Cylinders (L), J. Acoust. Soc. Am., vol 114, no. 1, pp. 45-47, Jul., 2003. [7] S. Robert et al., Resonant Elastic Scattering by a Finite Number of Cylindrical Cavities in an Elastic Matrix, Wave Motion, vol 40, pp. 225-239, 2004. [8] D. Yan, B. W. Drinkwater, and S. A. Neild, Measurement of the Ultrasonic Nonlinearity of Kissing Bonds in Adhesive Joints, NDT&E Int., vol. 42, pp. 459-466, 2009. [9] D. Yan, S. A. Neild, and B. W. Drinkwater, Modelling and Measurement of the Nonlinear Behaviour of Kissing Bonds in Adhesive Joints, NDT&E Int., vol. 45, pp. 18-25, 2012. [10] S. Biwa, S. Nakajima, and N. Ohno, On the Acoustic Nonlinearity of Solid-Solid Contact with Pressure-Dependent Interface Stiffness, J. Appl. Mech., vol 71, no. 4, pp. 508-515, Sep., 2004. [11] Y.-D. Feng, Y.-S, Wang, and A.-M. Zhang, Time domain BEM analysis of dynamic response of a cylinder embedded in soil with frictional slip at the interface, Soil Dynamics and Earthquake Engineering, vol 23, no. 4, pp. 303-311, 2003. [12] T. Sumiya, S. Biwa, and G. Haïat, Computational multiple scattering analysis of elastic waves in unidirectional composites, Wave Motion, vol 50, pp. 253-270, 2013. [13] K. Kawahara et al., Analysis of Second-Harmonic Generation Considering Nonlinear Mechanical Properties on Cylindrical Interface, Trans. Jpn. Soc. Mech. (in Japanese), vol 83, no. 849, 16-00582, 2017. [ID164] 7